Downing Street repeatedly refused to endorse the remarks of the Conservative chairman, which were criticised by some Tories.

Lady Warsi, the first Muslim woman to attend the Cabinet, sparked controversy when she declared that prejudice against Muslims had “passed the dinner table test” and was now seen as socially acceptable.

In remarks first reported by the Daily Telegraph, she also said that the practice of describing Muslims as either “moderate” or “extremist” fosters prejudice against all Muslims.

Mr Cameron’s official spokeswoman was asked repeatedly if the Prime Minister agreed with Lady Warsi, but refused to answer.

No 10 said: “The Prime Minister thinks this is an important debate. We want to see that debate continue. We’re looking forward to hearing what she has to say.”

It is also understood that, contrary to normal ministerial practice, Lady Warsi’s speech at Leicester University had not been cleared in advance with Downing Street.

Philip Hollobone, a Conservative MP who has called for a ban on Muslim women wearing the burkha in Britain, said Lady Warsi had not described the reality.

“One of the difficulties with Muslim communities is that a lot of people feel that they adapt least to our way of life. That is a perfectly legitimate concern to raise at the dinner table or anywhere else,” he said.

“There are other groups who are facing increasing prejudice in Britain today, not least Christians. It would be refreshing if Sayeeda Warsi came out to condemn that at the same time as condemning Islamophobia.”

Lord Tebbit, former Conservative chairman, suggested that Lady Warsi had been wrong to speak out.

“The Muslim faith was not discussed over the dinner tables of England, nor in the saloon bars, before large numbers of Muslims came here to our country. She might consider who is in need of her homilies on prejudice,” he said. “A period of silence from the baroness might not come amiss.”

Michael Nazir-Ali, a Church of England bishop, insisted that the distinction between extremist and moderate Muslims was valid.

He said: “We must distinguish between those Muslims who want to live peacefully with their non-Muslim neighbours and those who wish to introduce Shari’a into this country, restrict freedom of speech and confine women to their homes”

By contrast, Muslim commentators backed the minister’s approach.

Ghaffar Hussain of the Quilliam Foundation think tank said the speech would help address prejudice.

“While Islamist terrorism and Islamist extremism pose a clear danger to our society that needs to be tackled, this cannot justify the demonization of Muslims as a whole. British Muslims have a right to live their lives without fear of attack and without being discriminated against because of their religion.”

Shahid Mursaleen of the Minhaj-ul-Quran charity, also praised the speech.

“I agree Islam is misunderstood,” he said. “Islam is a peaceful and tolerant faith and I agree there is a real need to educate the public and to promote the truth about Islam.”

Following an outcry from Conservatives, Lady Warsi rewrote sections of the speech she had planned to deliver.

But in a BBC interview, she stood by her central argument about the spread of Islamophobia.

“It has seeped into our society in a way that it is acceptable to have these conversations where anti-Muslim hatred and bigotry is openly discussed,” she said.

“This is not about controlling the conversations that go on in people’s homes

This is about drawing a line as to the state of anti-religious hatred or bigotry in Britain today.”